Type IIB Flux Vacua via the String Worldsheet Jock McOrist University of Chicago William Linch, J.M. and Brenno Vallilo arXiv: 0804.0613
Motivation: Shortcomings and Expectations How to connect with reality? Flux compactifications are important � Approaches typically confined to SUGRA. Need large volume limit to control � String scale cycle: lots of string corrections SUGRA valid if ls << R What if a cycle approaches string scale? Need a string description! How will string theory change our SUGRA intuition? � Generic string solutions are expected to be string scale � Phenomenologically desirable to have no moduli -> hard in SUGRA � Are there new solutions not seen in SUGRA (eg. Non-geometric..?) Such new � solutions may be phenomenologically interesting 5/31/2008 Jock McOrist - String Phenomenology 2008 2
Type II String Theory Vacua Likely still have much to uncover beyond SUGRA Large volume flux Non-geometry,…? backgrounds Calabi-Yau Non-geometry,…? 5/31/2008 Jock McOrist - String Phenomenology 2008 3
How to Go Beyond Supergravity? Traditionally, string descriptions of RR fluxes are hard & not well � studied Three main approaches � Ramond-Neveu-Schwarz (RNS) � RR vertex operators have branch cuts, half integral picture, ... � Green-Schwarz (GS) � No covariant quantization � Light cone gauge is inconsistent for general flux vacua � D=4 Hybrid (Berkovits,…) � SO(3,1) covariant quantization � Circumvents the above problems nicely � Subjected to many tests (spectrum, scattering amplitudes … ) � Well suited to flux compactifications � 5/31/2008 Jock McOrist - String Phenomenology 2008 4
Outline Motivation: Why Do We Care About String Compactifications? 1. Some Lessons from Supergravity 2. Flux Vacua in the Hybrid 3. Physical Effects and Applications 4. Conclusions and Outlook 5. 5/31/2008 Jock McOrist - String Phenomenology 2008 5
Some Lessons from Supergravity: CY 3 Without fluxes: M 6 = CY 3 � Preserves N=2 spacetime supersymmetry. N = 2 � CY 3 Field content is given by KK reduction on the CY � Supergravity Multiplet � h 2,1 Vector Multiplets (complex structure moduli) � h 1,1 Hypermutiplets (Kahler moduli and dilaton) h 1;1 + 1 � 5/31/2008 Jock McOrist - String Phenomenology 2008 6
Some Lessons from Supergravity: Conformally CY 3 Simple Class of Solutions with G 3 = F 3 – τ H 3 � N = 1 Supersymmetry broken to N=1 � G 3 SUSY => G3 is (2,1) � Moduli lifted by: � Geometry backreacts: � Spacetime filling five-form related to warp factor � We study these backgrounds in string theory � Study non-compactly supported fluxes (evade tadpole, quantization) � Hence, work perturbatively in fluxes � SUGRA is also valid -> give us a concrete check of our approach � 5/31/2008 Jock McOrist - String Phenomenology 2008 7
Outline Motivation: Why Do We Care About String Compactifications? 1. Some Lessons from Supergravity 2. String Theory for RR Fluxes 3. Physical Effects and Applications 4. Conclusions and Outlook 5. 5/31/2008 Jock McOrist - String Phenomenology 2008 8
String Theory for Flux Vacua: D=4 Hybrid Hybrid originally formulated on with field content � Understandable as field redefinition of an N=1 critical RNS string: � GS a redefinition of RNS D=4 & ghost variables � usual RNS CY variables. Decoupled from D=4 sector. � Worldsheet Action � Comments: � Spacetime fermions have no branch cuts => manifest spacetime SUSY � Even though internal theory is RNS, we show how it can describe RR fluxes � described by (2,2) c = 9 SCFT � BRST + conformal invariance of N=1 RNS string <=> is a (2,2) SCFT. � => Entire worldsheet theory is a (2,2) SCFT. (2,2) worldsheet superconformal invariance required for theory to be physically � well-defined. We use it as our guiding principle 5/31/2008 Jock McOrist - String Phenomenology 2008 9
String Theory for Flux Vacua: Flux Vertex Operators Three-form fluxes. By KK reducing on the CY: � F_3 = � H_3 = � Map RNS vertex operators to Hybrid. Trick: internal fluxes correspond to � spacetime auxiliary fields For example: � p=1,…,h21 labels the (2,1) cohomology elements � Psi is RR ground state corresponding to pth cohomology element � O is the (c,c) element attained by spectral flow of � Vh has no branch cuts. May be integrated into the Hybrid action! � 5/31/2008 Jock McOrist - String Phenomenology 2008 10
String Theory for Flux Vacua: Flux Vertex Operators Have also written down vertex operators for other possible internal fluxes: � H : � where is the correction to the Levi-Civita connection, E is the complexified metric in a certain picture F1 and spacetime filling F5 : � G3 : � 5/31/2008 Jock McOrist - String Phenomenology 2008 11
Outline Motivation: Why Do We Care About String Compactifications? 1. Some Lessons from Supergravity 2. String Theory for RR Fluxes 3. Physical Effects and Applications 4. Conclusions and Outlook 5. 5/31/2008 Jock McOrist - String Phenomenology 2008 12
Physical Effects and Applications Start with non-compact Calabi-Yau background � Turn on small amount of Some expectations from Supergravity: � Spacetime becomes warped 1. Superpotential generated 2. We see each of these effects in string theory � Construct the integrated vertex operator for G3 flux � Deform worldsheet action by � As spacetime SUSY manifest, easy to see this breaks half the SUSY � corresponding to 5/31/2008 Jock McOrist - String Phenomenology 2008 13
Physical Effects and Applications: Warping & F 5 The deformed action is not conformally invariant at 1-loop in � The 1-loop beta function given by the UV structure of the two point � function: Divergence breaks conformal invariance To maintain conformal invariance we add a counter-term. Requiring D=4 � µ Poincare, the only vertex operator with the correct t structure is: The deformation preserves conformal invariance provided � Implies the spacetime metric has been adjusted to give precisely warping! � Similarly, one can show conformal invariance preserved only if we have � 5/31/2008 Jock McOrist - String Phenomenology 2008 14
Physical Effects and Applications: Superpotential Presence of flux => potential generated at tree level which lifts moduli � See this in string theory by tree-level scattering amplitude � = vertex operator for complex structure modulus In a CY 3 background, SL(2,C) and worldsheet SUSY => . � Flux background inserts a vertex operator, rendering the amplitude non- � zero Manifest spacetime SUSY => scattering amplitude automatically computes � a superpotential Topological Integration over zero modes invariant of CY and unbroken SUSY This may be recast in a more familiar form: � 5/31/2008 Jock McOrist - String Phenomenology 2008 15
Conclusions and Outlook Summary: � Important to have string theory description of flux vacua � This may lead to new string solutions, which are phenomenologically viable (no moduli, � string scale, understanding of the landscape…) We have shown how to identify flux vertex operators in the Hybrid � Computed using string theory effects well-known in supergravity � Future and Current Work: � Tip of the iceberg: many computable interesting physical effects. � Need to understand finite flux deformations and orientifolding (compact solutions) � Construction of a Hybrid GLSM ( work in progress with J. Park, C. Quigley, D. � Robbins, S. Sethi ) Eventually understand vacua that are string scale: eg. non-geometric, no volume � modulus. 5/31/2008 Jock McOrist - String Phenomenology 2008 16
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